For models fitted with betabin or quasibin, Pearson's residuals are computed as:
$$\frac{y - n * \hat{p}}{\sqrt{n * \hat{p} * (1 - \hat{p}) * (1 + (n - 1) * \hat{\phi})}}$$
where \(y\) and \(n\) are respectively the numerator and the denominator of the response, \(\hat{p}\)
is the fitted probability and \(\hat{\phi}\) is the fitted overdispersion parameter. When \(n = 0\), the
residual is set to 0. Response residuals are computed as \(y/n - \hat{p}\).
For models fitted with negbin or quasipois, Pearson's residuals are computed as:
$$\frac{y - \hat{y}}{\sqrt{\hat{y} + \hat{\phi} * \hat{y}^2}}$$
where \(y\) and \(\hat{y}\) are the observed and fitted counts, respectively. Response residuals are
computed as \(y - \hat{y}\).
data(orob2)
fm <- betabin(cbind(y, n - y) ~ seed, ~ 1,
link = "logit", data = orob2)
#Pearson's chi-squared goodness-of-fit statisticsum(residuals(fm, type = "pearson")^2)